Paired Sampling in Density-Sensitive Active Learning Pinar Donmez joint work with Jaime G. Carbonell Language Technologies Institute School of Computer Science Carnegie Mellon University

Outline Problem setting Motivation Our approach Experiments Conclusion

Setting X: feature space, label set Y={-1,+1} Data D ~ X x Y D = T U U  T: training set U: unlabeled set T is small initially, U is large Active Learning:  Choose most informative samples to label  Goal: high performance with least number of labeling requests

Motivation Optimize the decision boundary placement  Sampling disproportionately on one side may not be optimal  Maximize likelihood of straddling the boundary with paired samples Three factors affect sampling  Local density  Conditional entropy maximization  Utility score

Illustrative Example Left Figure  significant shift in the current hypothesis  large reduction in version space Right Figure  small shift in the current hypothesis  small reduction in version space Paired sampling Single point sampling

Density-Sensitive Distance Cluster Hypothesis:  decision boundary should NOT cut clusters  squeeze distances in high density regions  increase distances in low density regions Solution: Density-Sensitive Distance  find the weakest link along each path in a graph G  a better way to avoid outliers (i.e. a very short edge in a long path) Chapelle & Zien (2005)

Density-Sensitive Distance Apply MDS (Multi-dimensional Scaling) to to obtain a Euclidean embedding Find eigenvalues and eigenvectors of Pick the first p eigenvectors s.t.

Active Sampling Procedure Given a training set T in MDS space 1.Train logistic regression classifier on T 2.For all  Compute the pairwise score 3.Choose the pair with the maximum score 4.Repeat 1-3

Details of the Scoring Function S Two components of S 1.Likelihood of a pair having opposite labels (straddling the decision boundary) 2.Utility of the pair  By cluster assumption  decision boundary should not clusters => points in different clusters are likely to have different labels  In the transformed space, points in different clusters have low similarity (large distance) Thus, we can estimate

An Analysis Justifying our Claim Pairwise distances are divided into bins Pairs are assigned to bins acc. to their distances For each bin, relative frequency of pairs with opposite class labels are computed This graph (empirically) shows that likelihood of having opposite labels for two points monotonically increases with the pairwise distance between them. * This graph is plotted on g50c dataset.

Utility Function Two components  Local density depends on number of close neighbors their proximity  Conditional Entropy  For binary problems

Uncertainty-Weighed Density captures  the density of a given point  information content of its neighbors novelty:  each neighbor’s contribution weighed by its uncertainty  reduces the effect of highly certain neighbors  dense points with highly uncertain neighbors become important

Utility Function utility of a pair is regularize  information content (entropy) of the pair  proximity-weighted information content of neighbors

Experimental Data pair with maximum score selected Six binary datasets

Experiment Setting For each data set  start with 2 labeled data points (1 +, 1 -)  run each method for 20 iterations  results averaged over 10 runs Baselines  Uncertainty Sampling  Density-only Sampling  Representative Sampling (Xu et. al. 2003)  Random Sampling

Results

Conclusion Our contributions:  combine uncertainty, density, and dissimilarity across decision boundary  proximity-weighted conditional entropy selection is effective for active learning Results show  our method significantly outperforms baselines in error reduction fewer labeling requests than others to achieve the same performance

Thank You!